Changing the domain for a site

I have a client whose domain address is https://www.redlandcityliving.com.au and she wants to change the site to https://www.middleagedmama.com.au.

She has purchased the new domain and I currently have it parked on the same hosting space as the original domain. So, when you enter https://middleagedmama.com.au/ you get the site showing up under that domain.

To change all the references throughout the site do I have to make changes in the database or anywhere else or can I simply make the changes in the General Settings re the domain?

And is there a plugin that will automatically update all references to the new domain?

Looking for a simple way to make this change. Thanks.